Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 7 Fourier Series (Fourier 급수)
Lecture1 Periodic function
Lecturer: Lee, Yunsang (Physics)
Baird-Hall 01318
ylee@ssu.ac.kr
02-820-0404
1
1. Introduction
Problems involving vibrations or oscillations occur frequently in physics
and engineering. You can think of examples you have already met: a vibrating
tuning fork, a pendulum, a weight attached to a spring, water waves, sound waves,
alternating electric currents, etc. In addition, there are many more examples which
you will meet as you continue to study physics. On the other hand, Some of them –
for example, heat conduction, electric and magnetic fields, light – do not appear in
elementary work to have anything oscillatory about them, but will turn out in your
advanced work to involve the sine and cosines which are used in describing simple
harmonic motion and wave motion.
 It is why we learn how to expand a certain function with Fourier series
consisting of ‘infinite’ sines and cosines.
2
2. Simple harmonic motion and wave motion: periodic functions
(단순 조화운동과 파동운동 ; 주기함수)
1) Harmonic motion (단순 조화 운동)
- P moves at constant speed around a circle of radius A.
- Q moves up and down in such a way that its y coordinate is always equal to that
of P.
The back and forth motion of Q
 simple harmonic motion
For a constant circular motion,
2
  t ,
 (
 2f ) : angular ve locity
T
y coordinate of Q (or P): y  A sin   A sin t
3
2) Using complex number (복소수의 사용)
The x and y coordinates of P:
x  A cost , y  A sin t
Then, it is often convenient to use the complex notation.
In the complex plane, z  x  iy  A(cost  i sin t )
 Aeit
(Position of Q: imaginary part of the complex z)
dz d
 ( Ae it )  Ai eit  Ai  (cos t  i sin t )
Velocity:
dt dt
imaginary part  velocity of Q
4
3) Periodic function (함수의 주기)
i) Functional form of the simple harmonic motion:
A sin t or A cost , A sin(t   )
cf. phase difference or different choice of the origin
Displacement
Time
5
ii) Graph
a. Time (simple harmonic motion)
period: T 
Displacement
2

Time
amplitude
y  A sin t
dy
 A cost  B cost
dt
2
1  dy  1
Kinetic energy:
m   mB2 cos2 t
2  dt  2
Total energy
1
2
2 2
2 2
(kinetic+ potential = max of kinetic E) = mB  A   A f
2
6
b. Distance (wave)
y  A sin
2x
y  A sin

2
2v 
 2
( x  vt)  A sin  x 
t

 

distance
Wavelength: λ
cf . T 

v
,
f 
1
T
c. Arbitrary periodic function (like wave)
f ( x  p)  f ( x)
p : period(or wavelength)
7
3. Applications of Fourier Series (Fourier 급수의 응용)
- Fundamental (first order): sin t , cost
- Higher harmonics (higher order):
sin(nt ), cos(nt )
- Combination of the fundamental and the harmonics  complicated periodic
function. Conversely, a complicated periodic function  the combination of the
fundamental and the harmonics (Fourier Series expansion).
8
ex) Periodic function
-What a-c frequencies (harmonics) make up a given signal and in what proportions?
 We can answer the above question by expanding these various periodic
functions with Fourier Series.
9
1
sin x  sin 2 x  sin 3x 
3
sin x, sin 2 x, sin 3x
0
0
0
5
10
15
20
0
5
10
15
20
sin x
0
0
5
10
15
20
1
sin x  sin 2 x 
2
0
Intensity
0
5
10
15
20
0
0
5
10
15
20
1
sin x  sin 2 x  sin 3x 
3
1
sin x  sin 2 x  sin 3x    sin 10 x 
10
0
0
5
10
15
20
10
4. Average value of a function
(함수의 평균값)
1) average value of a function
With the interval x 
ba
n
' Approximate' averageof f ( x) on (a, b) 
f ( x1 )  f ( x2 )  f ( x3 )    f ( xn )  f ( x1 )  f ( x2 )  f ( x3 )    f ( xn )x

n
nx
When nx  b  a, n  , & x  0, (concept of integration)

Averageof f ( x) on (a, b) 
b
a
f ( x)dx
ba
11
2) Average of sinusoidal functions (사인함수의 평균)




sin 2 xdx   cos2 xdx.

Similarly (for n  0),




sin 2 nxdx   cos2 nxdx

Using sin nx  cos nx  1,
2

2
  sin


2


nx  cos nx dx   dx  2
2


  sin nxdx   cos2 nxdx  

2

Averagevalue (overa period)of sin 2 nx (or cos2 nx)
1  2
1 
1
2
2
2

sin
nxdx

cos
nxdx

cf
.
cos
2
x

cos
x

sin
x


2 
2 
2
sin 2 x 
1  cos2 x
1  cos2 x
, cos2 x 
2
2
12
Graph of sin2 nx
1.0
0.5
sin2x
0.0

-0.5
1.0
0.5
sin22x
0.0

-0.5
1.0
0.5
sin23x
0.0
-0.5
1.0

sin24x
0.5
0.0
-0.5
1.0

sin25x
0.5
0.0

-0.5
13
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 7 Fourier Series
Lecture 2 Basic of Fourier series
Lecturer: Lee, Yunsang (Physics)
Baird-Hall 01318
Email: ylee@ssu.ac.kr
Tel: 02-820-0404
14
5. Fourier coefficients (Fourier 계수)
We want to expand a given periodic function in a series of sines and cosines.
[First, we start with sin(nx) and cos(nx) instead of sin(nt) and cos(nt).]
- Given a function f(x) of period 2,
f ( x) 
1
a0  a1 cos x  a2 cos 2 x  a3 cos3x  
2
 b1 sin x  b2 sin 2 x  b3 sin 3x  ,
We need to determine the coefficients!!
15
In order to find formulas for an and bn, we need the following integrals on (-, )
1) Averagevalue of sin m xcosnx (overa period) (m, n : int eger)


1
2
 sin m xcosnxdx

1
2
1
 2 sin m  n x  sin m  n xdx  0
cf.



  sin nx dx  0

(n : int eger)
1
sin      sin    
2
1
cos sin   sin      sin    
2
1
cos cos   cos     cos   
2
1
sin  sin    cos     cos   
2
sin  cos  
16
2) Averagevalue of sin m xsin nx (overa period)

1

2
 sin m xsin nxdx

0, m  n

1  1
1








cos
m

n
x

cos
m

n
x
dx

 , mn0



2
2
2
0, m  n  0

cf.

  cosnx dx  0,

for interger n  0.
1
sin      sin    
2
1
cos sin   sin      sin    
2
1
cos cos   cos     cos   
2
1
sin  sin    cos     cos   
2
sin  cos  
17
3) Averagevalue of cosm xcosnx (overa period)

1
2

 cosm xcosnxdx

0, m  n


1
1
1







cos
m

n
x

cos
m

n
x
dx

 , mn0
2  2
2
1, m  n  0

1
sin      sin    
2
1
cos sin   sin      sin    
2
1
cos cos   cos     cos   
2
1
sin  sin    cos     cos   
2
sin  cos  
18
Using the above integrals, we can find coefficients of Fourier series by calculating
the average value.
f ( x) 
1
a0  a1 cos x  a2 cos 2 x  a3 cos3x  
2
 b1 sin x  b2 sin 2 x  b3 sin 3x  ,
i-1) To find a_o, we calculate the average on (-,)
1
2



f ( x)dx 
a0 1
2 2
1
 b1
2

1





dx  a1
1
2



1
sin
xdx

b
2

2
cos xdx  a2
 1

 2
1
2







 cos2 xdx  


 sin 2 xdx  


  f ( x)dx  a

0
19
i-2) To find a_1, we multiply cos x (n=1) and calculate the average on (-,).
 1

 2
1
2



1

a0 1 
1 
1
f ( x) cos xdx 
cos
xdx

a
cos
x
cos
xdx

a
1
2
2 2 
2 
2
1 
1 
 b1
sin
x
cos
xdx

b
sin 2 x cos xdx  
2






2
2

cos
x
 


 cos2 x cos xdx  


f ( x) cos xdx  a .




1
0, m  n


1
1
cf .
cos
m
x
cos
nxdx

 , mn0



2
2
1, m  n  0

20
i-3) To find a_2, we multiply cos 2x (n=2) and calculate the average on (-,).
 1

 2
1
2



1

a0 1 
1 
1
f ( x) cos2 xdx 
cos
2
xdx

a
cos
x
cos
2
dx

a
1
2
2 2 
2 
2
1 
1 
 b1
sin
x
cos
2
xdx

b
sin 2 x cos2 xdx  
2






2
2


cos
2
x






cos2 2 xdx  

f ( x) cos2 xdx  a .




2
0, m  n


1
1
cf .
cos
m
x
cos
nxdx

 , mn0



2
2
1, m  n  0

21
i-4) To find a_n, we multiply cos nx and calculate the average on (-,).
 1

 2
1
2



1


cos
nx




1 
2
f ( x) cosnxdx    an
cos
nxdx  
2 
1 
 b1
sin x cosnxdx  



2

f ( x) cosnxdx  a .




n
0, m  n


1
1
cf .
cos
m
x
cos
nxdx

 , mn0



2
2
1, m  n  0

22
ii-1) To find b_1 and b_n, (cf. n=0 term is zero), we multiply the sin x (n=1) or sin
nx and calculate the average on (-,).
 1 


 2
1
2



f ( x) sin xdx 
 b1
1

2



1
2
a0 1
2 2






sin xdx  a1
sin 2 xdx  b2
1
2
1
2



cos x sin dx  a2
1
2
 sin nx


 cos2 x sin xdx  


 sin 2 x sin xdx  

1
f ( x) sin xdx  b1.
2
Similarly,
1


  f ( x) sin nxdx  b

n
0, m  n


1
1
cf .
sin
m
x
sin
nxdx

 , mn0



2
2
0, m  n  0

23
## Fourier series expansion
1
f ( x)  a0  a1 cos x  a2 cos2 x  a3 cos3x  
2
 b1 sin x  b2 sin 2 x  b3 sin 3x  .
 an 
1


  f x cosnxdx,

bn 
1


  f x sin nxdx

.
24
0,    x  0,
f ( x)  
1, 0  x   .
Example 1.
an 
1

f ( x) cosnxdx





1 0

0  cosnxdx   1  cosnxdx


0
  
1


1

  cosnxdx  
 0
1



1
 sin nx  0
n
0
for n  0
  1
for n  0.
25
bn 



1
f ( x) sin nxdx





1 0
0

sin
nxdx

1  sin nxdx



0
  
1


0
0

 2

 n
 f ( x) 



1   cosnx 
1
n
sin nxdx  


(

1
)
1

  n 0
n
for even n
for odd n.
1 2  sin x sin 3x sin 5 x

 


 .
2  1
3
5

26
0,    x  0,
f ( x)  
1, 0  x   .
f ( x) 
1 2  sin x sin 3x sin 5 x

 


 .
2  1
3
5

9% overshoot
: Gibbs phenomenon
27
Example 2.
- case i
1
g x   2 f x   1
f ( x) 
1 2  sin x sin 3x sin 5 x

 


 .
2  1
3
5


4  sin x sin 3x sin 5 x



 

 1
3
5

- case ii
1


h x   f  x  
2











1 2  sin x  2 sin 3 x  2 sin 5 x  2
 


 

2 
1
3
5



1 2  cos x cos3x cos5 x

 


 
2  1
3
5

28
6. Dirichlet conditions (Dirichlet 조건)
: convergence problem (수렴 문제)
Does a Fourier series converge or does it converge to the values of f(x)?
-Theorem of Dirichlet:
If f(x) is
1) periodic of period 2
2) single valued between -  and 
3) a finite number of Max., Min., and discontinuities
4) integral of absolute f(x) is finite,
then,
1) the Fourier series converges to f(x) at all points where f(x) is continuous.
2) at jumps (e.g. discontinuity points), converges to the mid-point of the jump.
29
7. Complex form of Fourier series (Fourier 급수의 복소수 형태)
einx  e inx
sin nx 
,
2i
einx  e inx
cos nx 
.
2
Using these relations, we can get a series of terms of the forms e^inx and e^-inx
from the forms of sin nx and cos nx.
30
f ( x) 
1
a0  a1 cos x  a2 cos2 x  a3 cos3 x  
2
 b1 sin x  b2 sin 2 x  b3 sin 3x  
 eix  e  ix 
 e 2ix  e  2ix 
 e3ix  e  3ix 
1
  a2 
  a3 
  
 a0  a1 
2
2
2
2






 eix  e  ix 
 e 2ix  e  2ix 
 e3ix  e  3ix 
  b2 
  b3 
  
 b1 
2i
2i
 2i 





1
a b 
a b 
a b 
a b 
a0   1  1 eix   1  1 e  ix   2  2 e 2ix   2  2 e  2ix  
2
 2 2i 
 2 2i 
 2 2i 
 2 2i 
 c0  c1eix  c1e  ix  c2e 2ix  c 2e 2ix  

n
c e
n  
inx
n
31
f ( x) 
n
c e
n  
Here,
inx
n
cn 
1
2
.



f ( x)e  inx dx






  eimx einx dx   ei ( m  n ) x dx   cos(m  n) x  i sin(m  n) xdx
T hisintegralis non - zero only if m   n cosm  n x  0.
cf . an 
1

 

f ( x) cos nxdx ,
bn 
1

f ( x) sin nxdx




32
Example.
0,    x  0,
f ( x)  
1, 0  x   .
Expanding f(x) with the e^inx series,
1
cn 
2
0


e
inx
1
 0  dx 
2


o
e inx 1 dx
 1
,
1


e in  1  in
 2in
0,
0
1 
1 
1


c0 
f
x
dx

dx

.
2 
2 0
2
1 e inx

2  in



1
cn 
2



f ( x)e inx dx
n odd
n even  0,
33
Then,

f ( x)   cne

inx
1 1
 
2 i
 eix e3ix e5ix
 1
 

  
3
5
 1
 i
 e ix e 3ix e 5ix




 
5
 1  3

(convertingwith sinusoidal functions)
 1 2
1 2  eix  e  ix 1 e3ix  e  3ix
1

  

     sin x  sin 3x  
2   2i
3
2i
3

 2 
The same with the results of Fourier series with sines and cosines!!
34
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 7 Fourier Series
Lecture 3 Fourier series
Lecturer: Lee, Yunsang (Physics)
Baird-Hall 01318
ylee@ssu.ac.kr
02-820-0404
35
8. Other intervals (그 밖의 구간)
1) (-, ) and (0, 2 ).
- Same Fourier coefficients for the interval (-, ) and (0, 2 ).
1 
1 2
an   f ( x) cos nxdx   f ( x) cos nxdx
 
 0
1 
1 2
bn   f ( x) sin nxdx   f ( x) sin nxdx
 
 0
1 
1 2
inx
inx
cn 
f
(
x
)
e
dx

f
(
x
)
e
dx




0
2
2
(P roof)

0



0
  f ( x) cosnxdx    f ( x) cosnxdx 

2
0

f ( x) cosnxdx   f ( x) cosnxdx  

0
0
Here,  f ( x) cosnxdx  

2
2

f ( x) cosnxdx
f ( x) cosnxdx
f ( x) cosnxdx
 f ( x), cosnx : periodicfunction with 2 .
36
(Caution)
Different periodic functions made from the same function,
- same function f(x) = x^2
- different periodic with respect to the intervals, (-, ) and (0, 2 ).
37
2) period 2 vs. 2l
- Other period 2l [(0, 2l) or (-l, l)], not 2 [(0, 2) ]
sin nx  2
nx
nx inx / l
sin
(or cos
,e
)  2l
l
l
1 
1 l

2 
2l l
38
For f(x) with period 2l,
i) sinusoidal
a0
x
2x
f ( x)   a1 cos  a2 cos

2
l
l
a0  
nx
nx 
    an cos
 bn sin
.
2
l
l 
1 
1 l
nx
an   f ( x) cos
dx ,
l l
l
 b1 sin
x
l
 b2 sin
2x

l
1 l
nx
bn   f ( x) sin
dx.
l l
l
ii) complex

f ( x)   c n einx / l .

cn 
1 l
inx / l
f
(
x
)
e
dx.
2l l
39
Example.
0, 0  x  l ,
f ( x)  
1, l  x  2l
- period 2l
Using the complex functions as Fourier series,
1 2l
inx / l


f
x
e
dx

0
2l
1 l
1 2l inx / l
  0  dx   1  e
dx
0
l
2l
2l
cn 

inx / l
1 e
2l  in / l
2l

l

1
e  2in  e in
 2in

0, even n  0,
1


1  e in   1
 2in
  in , odd n,


 c0 
1 2l
1
dx

.

l
2l
2
40
Then,
1 1  ix / l ix / l 1 3ix / l 1 3ix / l

 e
e
 e
 e
 
2 i 
3
3

1 2  x 1
3x

   sin  sin
 .
2 
l 3
l

f ( x) 
41
9. Even and odd functions (짝함수, 홀함수)
1) definition
f ( x) is even if f ( x)  f ( x)
f ( x) is odd if f ( x)   f ( x)
42
- Even powers of x  even function, and odd powers of x  odd function.
- Any functions can be written as the sum of an even function and an odd function.
f ( x) 
1
2
1
 f ( x)  f ( x)  1  f ( x)  f ( x)
2
2
1
2
x
x
x
x
x
ex. e  (e  e )  (e  e )  cosh x (even)  sinh x (odd)
43
2) Integration
Integral over symmetric intervals like (-, ) or (-l, l)

l
l
0
if f ( x) is odd,


f ( x)dx   l
2 f ( x)dx if f ( x) is even.

 0
44
- In order to represent a f(x) on interval (0, l) by Fourier series of period 2l,
we need to have f(x) defined on (-l, 0), too.
- We can expand the function on (-l, 0) to be even or odd on (-l, 0).
Anything is OK!!
45
3) Fourier series
a0  
nx
nx 
f ( x)     an cos
 bn sin

2
l
l 
1 
- Cosine function: even,
Sine function: odd.
- If f(x) is even,
the terms in Fourier series should be even.  b_n should be zero.
- If f(x) is odd,
the terms in Fourier series should be odd.  a_n should be zero.
an  0

If f ( x) is odd, 
2 l
nx
b

f
(
x
)
sin
dx.
n


0
l
l

2 l
nx

a

f
(
x
)
cos
dx,
 n

0
If f ( x) is even, 
l
l

bn  0.
46
- How to represent a function on (0, 1) by Fourier series
1) sine-cosine or exponential (ordinary method) (period 1, l=1/2)
2) odd or even function (period 2, l=1)
(caution) different period!!
47
Example
1,
f ( x)  
0,
0  x  12
1
2
 x 1
(a) odd function (period 2)
 Fourier sine series.
(b) even function (period 2)
 Fourier cosine series.
(c) original function (period 1)
 Ordinary sine-cosine,
or exponential
48
(a) Fourier sine series (using odd function with period 2, l = 1)
an  0

If f ( x) is odd, 
2 l
nx
b

f
(
x
)
sin
dx.
n

l 0
l

1/ 2
2 1
bn   f ( x) sin nxdx  2  sin nxdx
0
l 0

2
2 
n
1/ 2

cosnx 0  
cos
 1,

n
n 
2

n


cos

1


1
,

2
,

1
,
0
,

for
n

1
,
2
,
3
,
4
,



2
b1 
2

f ( x) 
, b2 
4
2
0
, b3 
, b4 
,
2
3
4
2
2 sin 2x sin 3x sin 5x 2 sin 6x

sin

x




 


2
3
5
6

49
(b) Fourier sine series (using odd function with period 2, l = 1)
2 l
nx

a

f
(
x
)
cos
dx,
 n

0
If f ( x) is even, 
l
l

bn  0.
1
1/ 2
0
0
a 0  2 f ( x)dx  2 dx  1,
2
2
n
1/ 2
sin nx 0 
sin
.
0
n
n
2
1 2  cosx cos3x cos5x

f ( x)   


 
2  1
3
5

1
an  2 f ( x) cos nxdx 
50
(c) Ordinary Fourier series
i) exponential
1
1/ 2
0
0
cn   f ( x)e  2inx dx  
e  2inx dx
 1
1 e
1  (1)
 in ,



 2in
 2in
0,

 in
c0  
1/ 2
0
n
n odd,
n even.
dx  12 .
1 1 2in
 (e
 e  2in  13 e 6in  13 e 6in  )
2 i
1 2
sin 6x
  (sin 2x 
 ).
2 
3
f ( x) 
51
ii) sine-cosine
1
1/ 2
0
0
a0  2 f ( x)dx  2 dx  1
1/ 2
an  2 cos 2nxdx  0.
0
1/ 2
0
b1 
2

,
b 2  0,


1
1
(1  cos n ) 
1  (1) n .
n
n
2
b3 
, b 4  0,
3
bn  2 sin 2nxdx 
52
10. Application to sound (소리에 대한 응용)
- odd function
- period = 1/262
1 / 524
bn  2(524) 
0
p(t ) 
1
4
p(t ) sin 524ntdt   
2  15
n
7

 1  cosn 
  cos
n  8
2
8

 sin 524t 30sin(524 2t ) sin(524 3t )



1
2
3


sin(524 5t ) 30sin(524 6t ) sin(524 7t )



 
5
6
7

53
- Intensity of a sound wave is proportional to the average of the square of
amplitude, A2.
n
=
1
2
3
4
5
6
7
8
9
10
relative
=
intensity
1
225
1/9
0
1/25
25
1/49
0
1/81
9
- Second harmonics is dominant!!
54
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 7 Fourier Series
Lecture 4 Fourier Transform
Lecturer: Lee, Yunsang (Physics)
Baird-Hall 01318
ylee@ssu.ac.kr
02-820-0404
55
11. Parseval’s theorem (completeness relation)
(Parseval’s 정리 ; 완전성 관계)

f ( x)  a0   an cos nx  bn sin nx
1
2
1
Averageof  f ( x) 
2
1
2
2


f
(
x
)
dx  ?


Hint
1) average of (1/2a_0)^2 = (1/2a_0)^2
2) average of (a_n cos nx)^2) = a_n^2* 1/2
3) average of (b_n sin nx)^2 = b_n^2 * 1/2.
4) average of all cross product terms, a_n*b_m*cos nx*sin mx, = 0.
Averageof f ( x)   a0 
2
1
2
2
1  2 1  2
  an   bn
2 1
2 1

Similarly, averageof f ( x)   cn
2
2

- Parseval’s theorem or completeness relation
- The set of cos nx and sin nx is a complete set!!
56
12. Fourier Transforms (Fourier 변환)
- Periodic function  Fourier series with discrete frequencies

f x    cneinx / l ,
1 l
cn   f x einx / l dx
2l l

(Fourier series expansion)
- What happens for ‘non-periodic function’?
 Fourier transform with continuous frequencies

f x    g  e d ,
i x

1
g   
2



f x e  ix dx.
(Fourier t ransform)
cf. Fourier series vs. Fourier transform







d ,
n
 ,
l
1 l
1
dx

2l l
2



d
57
- Conversion of the Fourier series to the Fourier transform

f x    cnei n x , (

cn 
n

  n ,   )
l
l
1 l

 i n x


f
x
e
dx

2l  l
2
 
f x    
   2



  2


l
l

l
l

f u e i n u du. ( to avoid the confusion)
l
l

f u e  i n u duei n x

f u e
 i n  x  u 
1
du 
2

 F   ,

n
where F  n    f u ei n  x  u du.
l
l
F     f u ei  x  u du
l
l
f x  
1
2
1
g   
2






F  d 
f x e
 i x
1
2

 

 
1
dx 
2



f u ei  x  u dud 
1
2


e  ix d 



f u e  iu du.
f u e  iu du,

f x    g  eix d .

58
- Fourier sine/cosine transforms
e ix  cosx  i sin x
1
g   
2



f x cosx  i sin x dx
For odd f x ,
g   
1
2



f x  i sin x dx  

f x sin xdx.


i
0
g      g  ,  odd g  .
Similarly,


f x    g  e dx  2i  g  sin xd .

i x
0
59
i) Fourier Sine T ransform,
f s x  
g s x  
2



0
g s  sin xd ,

f  x sin xdx.


2
0
s
ii) Fourier Cosine T ransform,

f c x  

g c x  
f  x cosxdx.


2
0
2
g c  cosxd ,

0
c
60
Example 1.

1,  1  x  1,
f x   

0, x  1,
g   
1
2
1

2



f x e  ix dx
1 e  i x
 ix
1e dx  2  i
1

sin 


f x   
eix dx 
1
1
1 e  i  ei sin 


.
  2i

1

sin  cos x  i sin x 



dx 
2


0
sin  cos x

d
61
Example 2.

1,  1  x  1,
f x   

0, x  1,

 sin  cosx

2


d


f
x


0

2
0

For x  0,


0
sin 

d 

2
for x  1,

4
for x  1
for x  1.
.
62
- Parseval’s Theorem for Fourier integrals
1
g~1   
2



~
f1  x eix dx.

~  g  d  1
g
 1 2
2
1
2

~
ix

 1




f
x
dx
g

e
d

 1  2
 2


~  g a d  1
g
 1 2
2



i x
  ~


f
x
e
dx g 2  d .
  1


g   d 
2
1
2









~
f1  x  f 2 x dx
~
f1  x  f 2 x dx.
f  x  dx.
2

cf. averageof f ( x)   cn
2
2

63
- Various Fourier transforms
64
- Michelson interferometer
65
HW
Chapter 7
2-3, 9, 13, 18 (G1)
5-1, 7 (G2)
7-1 (G3)
9-1,6,7 (G4)
66